An Isaac Newton Institute Programme

Model Theory and Applications to Algebra and Analysis

Betti numbers of sets defined by quantifier-free formulas

Abstract

(Joint work with N. Vorobjov)

Upper bounds for the Betti numbers of real algebraic sets
were obtained by Oleinik-Petrovskii (1949), Milnor (1964) and
Thom (1965). These bounds, based on Morse theory, were single
exponential in the number of variables.
Basu (1999) extended these results to real semialgebraic sets
defined by equations and non-strict inequalities.
However, the best previously known upper bounds for general
semialgebraic sets were double exponential.
Gabrielov and Vorobjov (2005) obtained a single exponential
upper bound on the Betti numbers of a general semialgebraic set.
Given a semialgebraic set X, another semialgebraic set Y defined by
equations and non-strict inequalities is constructed,
with the same Betti numbers as X. Basu's theorem applied to Y
provides the upper bound for the Betti numbers of X.
The method easily generalizes to non-algebraic functions,
such as Pfaffian functions.